Polynomial Functions - ACT Math
Card 1 of 30
Identify the degree of $f(x)=7x^5-3x^2+9$.
Identify the degree of $f(x)=7x^5-3x^2+9$.
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Degree $=5$. The highest power term determines the degree.
Degree $=5$. The highest power term determines the degree.
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What does $\Delta=b^2-4ac<0$ imply about the roots of $ax^2+bx+c$?
What does $\Delta=b^2-4ac<0$ imply about the roots of $ax^2+bx+c$?
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No real roots (two complex roots). Negative discriminant means the parabola doesn't touch the $x$-axis.
No real roots (two complex roots). Negative discriminant means the parabola doesn't touch the $x$-axis.
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Identify the form: $ax^2 + bx + c$.
Identify the form: $ax^2 + bx + c$.
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Quadratic form. Standard form of a degree-2 polynomial.
Quadratic form. Standard form of a degree-2 polynomial.
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Find the sum of $2x^3 + x^2$ and $3x^3 - 5x^2$.
Find the sum of $2x^3 + x^2$ and $3x^3 - 5x^2$.
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$5x^3 - 4x^2$. Combine like terms: $(2+3)x^3 + (1-5)x^2$.
$5x^3 - 4x^2$. Combine like terms: $(2+3)x^3 + (1-5)x^2$.
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Subtract $4x + 7$ from $6x + 3$.
Subtract $4x + 7$ from $6x + 3$.
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$2x - 4$. $(6x + 3) - (4x + 7) = 2x - 4$.
$2x - 4$. $(6x + 3) - (4x + 7) = 2x - 4$.
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What is the product of $x + 2$ and $x - 3$?
What is the product of $x + 2$ and $x - 3$?
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$x^2 - x - 6$. Use FOIL: $x^2 + 2x - 3x - 6$.
$x^2 - x - 6$. Use FOIL: $x^2 + 2x - 3x - 6$.
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What is the degree of the polynomial $3x^4 + 5x^2 - 7$?
What is the degree of the polynomial $3x^4 + 5x^2 - 7$?
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The degree is 4. The highest power of $x$ is 4.
The degree is 4. The highest power of $x$ is 4.
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Identify the leading coefficient of $7x^5 - 3x^3 + x$.
Identify the leading coefficient of $7x^5 - 3x^3 + x$.
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The leading coefficient is 7. The coefficient of the highest degree term.
The leading coefficient is 7. The coefficient of the highest degree term.
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State the term for a polynomial with two terms.
State the term for a polynomial with two terms.
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Binomial. A polynomial with exactly two terms.
Binomial. A polynomial with exactly two terms.
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What is the term for a polynomial with one term?
What is the term for a polynomial with one term?
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Monomial. A polynomial with exactly one term.
Monomial. A polynomial with exactly one term.
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What is the degree of the polynomial $5x^2y^3 + 3xy - 4$?
What is the degree of the polynomial $5x^2y^3 + 3xy - 4$?
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The degree is 5. The sum of exponents in $x^2y^3$ is $2+3=5$.
The degree is 5. The sum of exponents in $x^2y^3$ is $2+3=5$.
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Find the remainder when $f(x)=x^3-4x+1$ is divided by $(x-2)$.
Find the remainder when $f(x)=x^3-4x+1$ is divided by $(x-2)$.
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$f(2)=1$. By the Remainder Theorem, $f(2) = 8 - 8 + 1 = 1$.
$f(2)=1$. By the Remainder Theorem, $f(2) = 8 - 8 + 1 = 1$.
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What does $\Delta=b^2-4ac=0$ imply about the roots of $ax^2+bx+c$?
What does $\Delta=b^2-4ac=0$ imply about the roots of $ax^2+bx+c$?
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One real double root. Zero discriminant means the parabola touches the $x$-axis once.
One real double root. Zero discriminant means the parabola touches the $x$-axis once.
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Identify the degree of $f(x)=7x^5-3x^2+9$.
Identify the degree of $f(x)=7x^5-3x^2+9$.
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Degree $=5$. The highest power term determines the degree.
Degree $=5$. The highest power term determines the degree.
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Identify the leading coefficient of $f(x)=-4x^6+2x-1$.
Identify the leading coefficient of $f(x)=-4x^6+2x-1$.
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Leading coefficient $=-4$. The coefficient of the highest degree term is the leading coefficient.
Leading coefficient $=-4$. The coefficient of the highest degree term is the leading coefficient.
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Find the discriminant of $f(x)=x^2-6x+13$ and classify the real roots.
Find the discriminant of $f(x)=x^2-6x+13$ and classify the real roots.
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$\Delta=-16$, no real roots. Calculate $\Delta = 36 - 52 = -16 < 0$, so no real roots.
$\Delta=-16$, no real roots. Calculate $\Delta = 36 - 52 = -16 < 0$, so no real roots.
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Identify the other complex root if $3+2i$ is a root of a real-coefficient polynomial.
Identify the other complex root if $3+2i$ is a root of a real-coefficient polynomial.
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$3-2i$. Complex conjugate pairs occur in real-coefficient polynomials.
$3-2i$. Complex conjugate pairs occur in real-coefficient polynomials.
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Identify whether the graph crosses or touches at $x=1$ for $f(x)=(x-1)^4(x+2)$.
Identify whether the graph crosses or touches at $x=1$ for $f(x)=(x-1)^4(x+2)$.
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Touches and turns at $x=1$. Even multiplicity $(x-1)^4$ means the graph touches and turns.
Touches and turns at $x=1$. Even multiplicity $(x-1)^4$ means the graph touches and turns.
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What does $\Delta=b^2-4ac=0$ imply about the roots of $ax^2+bx+c$?
What does $\Delta=b^2-4ac=0$ imply about the roots of $ax^2+bx+c$?
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One real double root. Zero discriminant means the parabola touches the $x$-axis once.
One real double root. Zero discriminant means the parabola touches the $x$-axis once.
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Find $f(0)$ for $f(x)=3x^4-5x+12$.
Find $f(0)$ for $f(x)=3x^4-5x+12$.
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$f(0)=12$. Substitute $x = 0$ to get the constant term.
$f(0)=12$. Substitute $x = 0$ to get the constant term.
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What is the value of $f(0)$ for $f(x)=a_nx^n+\cdots+a_0$?
What is the value of $f(0)$ for $f(x)=a_nx^n+\cdots+a_0$?
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$f(0)=a_0$. Substituting $x = 0$ eliminates all terms except the constant.
$f(0)=a_0$. Substituting $x = 0$ eliminates all terms except the constant.
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What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is even and $a_n<0$?
What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is even and $a_n<0$?
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As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like behavior.
As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like behavior.
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What is the sum of the zeros of $ax^2+bx+c$ (counting multiplicity)?
What is the sum of the zeros of $ax^2+bx+c$ (counting multiplicity)?
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$r_1+r_2=-\frac{b}{a}$. Vieta's formula relates coefficients to sums of roots.
$r_1+r_2=-\frac{b}{a}$. Vieta's formula relates coefficients to sums of roots.
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What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n>0$?
What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n>0$?
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As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient goes from bottom-left to top-right.
As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient goes from bottom-left to top-right.
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What is the product of the zeros of $ax^2+bx+c$ (counting multiplicity)?
What is the product of the zeros of $ax^2+bx+c$ (counting multiplicity)?
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$r_1r_2=\frac{c}{a}$. Vieta's formula relates coefficients to products of roots.
$r_1r_2=\frac{c}{a}$. Vieta's formula relates coefficients to products of roots.
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What is the Remainder Theorem for dividing $f(x)$ by $(x-r)$?
What is the Remainder Theorem for dividing $f(x)$ by $(x-r)$?
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Remainder $=f(r)$. The Remainder Theorem evaluates the polynomial at the divisor's zero.
Remainder $=f(r)$. The Remainder Theorem evaluates the polynomial at the divisor's zero.
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Identify the form: $ax^2 + bx + c$.
Identify the form: $ax^2 + bx + c$.
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Quadratic form. Standard form of a degree-2 polynomial.
Quadratic form. Standard form of a degree-2 polynomial.
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What is the multiplicity rule for a factor $(x-r)^k$ of $f(x)$?
What is the multiplicity rule for a factor $(x-r)^k$ of $f(x)$?
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Zero $r$ has multiplicity $k$. The exponent of the factor equals the multiplicity of the zero.
Zero $r$ has multiplicity $k$. The exponent of the factor equals the multiplicity of the zero.
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What is the discriminant of $ax^2+bx+c$ used to classify real roots?
What is the discriminant of $ax^2+bx+c$ used to classify real roots?
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$\Delta=b^2-4ac$. The discriminant determines the nature of quadratic roots.
$\Delta=b^2-4ac$. The discriminant determines the nature of quadratic roots.
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What is the vertex $x$-coordinate of $f(x)=ax^2+bx+c$?
What is the vertex $x$-coordinate of $f(x)=ax^2+bx+c$?
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$x_v=-\frac{b}{2a}$. The vertex $x$-coordinate is the axis of symmetry.
$x_v=-\frac{b}{2a}$. The vertex $x$-coordinate is the axis of symmetry.
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